smc-6312-30-IQR and Outliers

Statistics, STD2 EQ-Bank 9 MC

A dataset has an interquartile range (IQR) of 18.

The upper quartile (Q₃) is 45.

What is the maximum value that would NOT be classified as an outlier?

Show Answers Only

$D$

Show Worked Solution

$1.5 \times \text{IQR} = 1.5 \times 18 = 27$

$\text{Upper boundary} = Q_3 + 27 = 45 + 27 = 72$

$\text{Values up to and including 72 are not outliers}$

$\text{Values above 72 are outliers}$

$\therefore\ \text{Maximum value that is not an outlier} = 72$

$\Rightarrow D$

Statistics, STD2 EQ-Bank 8 MC

The following data shows the ages of people attending a fitness class:

16, 24, 26, 28, 30, 32, 34, 36, 52, 62

How many outliers are in this dataset?

Show Answers Only

$C$

Show Worked Solution

$\text{Arrange in order (already ordered)}\ n = 10 $

$\text{Median }(Q_2) = \dfrac{30 + 32}{2} = 31 $

$Q_1 = 26$

$Q_3 =36$

$\text{IQR} = 36-26 = 10 $

$1.5 \times \text{IQR} = 1.5 \times 10 = 15$

$\text{Lower boundary} = 26-15 = 11 $

$\text{Upper boundary} = 36+15 = 51$

$\text{Values 52 and 62 are both above 51} \therefore\ 2 \text{ outliers}$

$\Rightarrow C$

Statistics, STD2 EQ-Bank 7 MC

A dataset has a lower quartile (Q₁) of 25 and an upper quartile (Q₃₎ of 35.

Which of the following values would be classified as an outlier?

Show Answers Only

$A$

Show Worked Solution

$\text{IQR}=Q_3-Q_1$

$=35-25=10$

$\text{Lower boundary}=25-1.5\times 10=10$

$\text{Upper boundary}=35+1.5\times 10=50$

$\text{Check each value:} $

$2 < 10 \text{ (outlier)} $

$10 = 10 \text{ (on boundary, not an outlier)}$

$48 < 50 \text{ (not an outlier)} $

$50 = 50 \text{ (on boundary, not an outlier)}$

$\therefore\ 2\ \text{is the only outlier}$

$\Rightarrow A$

Statistics, STD2 EQ-Bank 6 MC

A dataset has the following values arranged in order:

12, 15, 18, 21, 24, 27, 30, 33, 36

What is the interquartile range (IQR) of this dataset?

Show Answers Only

$B$

Show Worked Solution

$\text{Number of data values: }n=9$

$\text{Median }(Q_2): =24$

$\text{Lower half of data: }12, 15, 18, 21$

$Q_1=\dfrac{15+18}{2}\ =16.5$

$\text{Upper half of data: } 27, 30, 33, 36$

$Q_3=\dfrac{30+33}{2}\ =31.5$

$\text{IQR}=Q_3-Q_1$

$=31.5-16.5=15$

$\Rightarrow C$

Statistics, STD2 S1 2021 HSC 17

The five-number summary of a dataset is given.

Lowest score = 1

Lowest quartile (`Q_1`) = 4

Median (`Q_2`) = 7

Upper quartile (`Q_3`) = 10

Highest score = 20

Is 20 an outlier? Justify your answer with calculations. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(20 is an outlier – See Worked Solution)`

Show Worked Solution

`IQR = Q_3 – Q_1 = 10-4=6`

COMMENT: Stating 20 > 19 is necessary to justify this answer.

`text(Upper Fence)`	`= Q_3 + 1.5 xx IQR`
	`=10 + 1.5 xx 6`
	`=19`

`text{S}text{ince 20 > 19, 20 is an outlier.}`

Statistics, STD2 S1 2019 HSC 19

The heights, in centimetres, of 10 players on a basketball team are shown.

170, 180, 185, 188, 192, 193, 193, 194, 196, 202

Is the height of the shortest player on the team considered an outlier? Justify your answer with calculations. (3 marks)

Show Answers Only

`text(See Worked Solutions)`

Show Worked Solution

`Q_1 = 185, \ Q_3 = 194`

Mean mark 51%.
COMMENT: The last statement must be made to achieve full marks here!

`IQR = 194 – 185 = 9`

`text(Shortest player = 170)`

`text(Outlier height:)`

`Q_1 – 1.5 xx IQR `	`= 185 – 1.5 xx 9`
	`= 171.5`

`:.\ text(S)text(ince 170 < 171.5, 170 is an outlier.)`

Statistics, STD2 S1 2018 HSC 26e

A cumulative frequency table for a data set is shown.

\begin{array} {|c|c|}
\hline
\ \ \ \ \ \ \ \textit{Score}\ \ \ \ \ \ \ & \ \ \ \ \ \textit{Cumulative}\ \ \ \ \ \\ & \textit{frequency} \\
\hline
\rule{0pt}{2.5ex} \text{1} \rule[-1ex]{0pt}{0pt} & 5 \\
\hline
\rule{0pt}{2.5ex} \text{2} \rule[-1ex]{0pt}{0pt} & 9 \\
\hline
\rule{0pt}{2.5ex} \text{3} \rule[-1ex]{0pt}{0pt} & 16 \\
\hline
\rule{0pt}{2.5ex} \text{4} \rule[-1ex]{0pt}{0pt} & 20 \\
\hline
\rule{0pt}{2.5ex} \text{5} \rule[-1ex]{0pt}{0pt} & 34 \\
\hline
\rule{0pt}{2.5ex} \text{6} \rule[-1ex]{0pt}{0pt} & 42 \\
\hline
\end{array}

What is the interquartile range of this data set? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`2`

Show Worked Solution

`text(42 data points ⇒ median) = text(21st + 22nd)/2`

♦♦ Mean mark 27%.

`text(Q)_1`	`= 11text(th data point) = 3`
`text(Q)_3`	`= 32text(nd data point) = 5`

`:.\ text(IQR)`	`= 5 – 3`
	`= 2`

Statistics, STD2 S1 2017 HSC 30a

A set of data has a lower quartile (`Q_L`) of 10 and an upper quartile (`Q_U`) of 16.

What is the maximum possible range for this set of data if there are no outliers? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`24`

Show Worked Solution

`IQR = 16 – 10 = 6`

♦♦ Mean mark 34%.

`text(If no outliers,)`

`text(Upper limit)`	`= Q_U + 1.5 xx IQR`
	`= 16 + 1.5 xx 6`
	`= 25`

`text(Lower limit)`	`= Q_L – 1.5 xx IQR`
	`= 10 – 1.5 xx 6`
	`= 1`

`:.\ text(Maximum range)`	`= 25 – 1`
	`= 24`

Statistics, STD2 S1 2015 HSC 27d

In a small business, the seven employees earn the following wages per week:

$\$300, \ \$490, \ \$520, \ \$590, \ \$660, \ \$680, \ \$970$

Is the wage of $970 an outlier for this set of data? Justify your answer with calculations. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---
Each employee receives a $20 pay increase.

What effect will this have on the standard deviation? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $\text{See Worked Solutions.} $

b. $\text{The standard deviation will remain the same.}$

Show Worked Solution

a. $300, 490, 520, 590, 660, 680, 970$

$\text{Median}$	$= 590$
$Q_1$	$= 490$
$Q_3$	$= 680$
$IQR$	$= 680-490 = 190$

$\text{Outlier if \$970 is greater than:} $

$Q_3 + 1.5 x\times IQR = 680 + 1.5 \times 190 = \$965 $

$\therefore\ \text{The wage \$970 per week is an outlier.}$

♦ Mean mark (a) 39%.

b. $\text{All values increase by \$20, but so too does the mean.} $

$\text{Therefore the spread about the new mean will not change} $

$\text{and therefore the standard deviation will remain the same.} $

Statistics, STD2 S1 2007 HSC 21 MC

This set of data is arranged in order from smallest to largest.

`5, \ 6, \ 11, \ x, \ 13, \ 18, \ 25`

The range is six less than twice the value of `x`.

Which one of the following is true?

The median is 12 and the interquartile range is 7.
The median is 12 and the interquartile range is 12.
The median is 13 and the interquartile range is 7.
The median is 13 and the interquartile range is 12.

Show Answers Only

`D`

Show Worked Solution

`5, 6, 11, x, 13, 18, 25`

`text(Range)`	`= 2x – 6`
`25 – 5`	`= 2x – 6`
`2x`	`= 26`
`x`	`= 13`
`:.\ text(Median)`	`= 13`

`Q_1 = 6\ \ \ \ \ Q_3 = 18`

`:.\ text(IQR) = 12`

`=> D`

\(\text{Median}\)	\(= 590\)
\(Q_1\)	\(= 490\)
\(Q_3\)	\(= 680\)
\(IQR\)	\(= 680-490 = 190\)